Editing Homology sphere

{{Short description|Topological manifold whose homology coincides with that of a sphere}}
In [[algebraic topology]], a '''homology sphere''' is an ''n''-[[manifold]] ''X'' having the [[homology group]]s of an ''n''-[[sphere]], for some integer <math>n\ge 1</math>. That is,

:<math>H_0(X,\Z) = H_n(X,\Z) = \Z</math> 

and

:<math>H_i(X,\Z) = \{0\}</math> for all other ''i''.

Therefore ''X'' is a [[connected space]], with one non-zero higher [[Betti number]],  namely, <math>b_n=1</math>. It does not follow that ''X'' is [[simply connected]], only that its [[fundamental group]] is [[perfect group|perfect]] (see [[Hurewicz theorem]]).

A [[rational homology sphere]] is defined similarly but using homology with rational coefficients.

==Poincaré homology sphere==
<!-- Henri Poincaré links here -->
The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by [[Henri Poincaré]].  Being a [[spherical 3-manifold]], it is the only homology 3-sphere (besides the [[3-sphere]] itself) with a finite [[fundamental group]].  Its fundamental group is known as the [[binary icosahedral group]] and has order 120.  Since the fundamental group of the 3-sphere is trivial, this shows that there exist 3-manifolds with the same homology groups as the 3-sphere that are not homeomorphic to it.

===Construction===
A simple construction of this space begins with a [[dodecahedron]]. Each face of the dodecahedron is identified with its opposite face, using the minimal clockwise twist to line up the faces. [[Quotient space (topology)|Gluing]] each pair of opposite faces together using this identification yields a closed 3-manifold. (See [[Seifert–Weber space]] for a similar construction, using more "twist", that results in a [[hyperbolic 3-manifold]].)

Alternatively, the Poincaré homology sphere can be constructed as the [[Quotient space (topology)|quotient space]] [[SO(3)]]/I where I is the [[Icosahedral symmetry|icosahedral group]] (i.e., the rotational [[symmetry group]] of the regular [[icosahedron]] and dodecahedron, isomorphic to the [[alternating group]] A<sub>5</sub>). More intuitively, this means that the Poincaré homology sphere is the space of all geometrically distinguishable positions of an icosahedron (with fixed center and diameter) in Euclidean 3-space. One can also pass instead to the [[universal cover]] of SO(3) which can be realized as the group of unit [[quaternion]]s and is [[homeomorphic]] to the 3-sphere. In this case, the Poincaré homology sphere is isomorphic to <math>S^3/\widetilde{I}</math> where <math>\widetilde{I}</math> is the [[binary icosahedral group]], the perfect [[Double covering group|double cover]] of I [[Embedding|embedded]] in <math>S^3</math>.

Another approach is by [[Dehn surgery]]. The Poincaré homology sphere results from +1 surgery on the right-handed [[trefoil knot]].

===Cosmology===
In 2003, lack of structure on the largest scales (above 60 degrees) in the [[cosmic microwave background]] as observed for one year by the [[Wilkinson Microwave Anisotropy Probe|WMAP]] spacecraft led to the suggestion, by [[Jean-Pierre Luminet]] of the [[Observatoire de Paris]] and colleagues, that the [[shape of the universe]] is a Poincaré sphere.<ref name="physwebLum03">[https://physicsworld.com/a/is-the-universe-a-dodecahedron/ "Is the universe a dodecahedron?"], article at PhysicsWorld.</ref><ref name="Nat03">{{cite journal
  | last1 = Luminet
  | first1 = Jean-Pierre
  | author1-link = Jean-Pierre Luminet
  |author2-link=Jeffrey Weeks (mathematician)|first2=Jeff|last2= Weeks 
  |first3=Alain|last3= Riazuelo |first4=Roland|last4= Lehoucq 
  |first5=Jean-Phillipe |last5=Uzan
  | title = Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background
  | volume = 425
  | issue = 6958
  | pages = 593–595
  |journal = [[Nature (journal)|Nature]]
  | date = 2003-10-09
  | arxiv = astro-ph/0310253
  | issn = 
  | doi = 10.1038/nature01944
  | id = 
  | pmid = 14534579
  | bibcode=2003Natur.425..593L| s2cid = 4380713
 }}</ref> In 2008, astronomers found the best orientation on the sky for the model and confirmed some of the predictions of the model, using three years of observations by the WMAP spacecraft.<ref name="RBSG08">{{cite journal
  | last1 =Roukema
  | first1 =Boudewijn
  |first2=Zbigniew|last2= Buliński |first3=Agnieszka|last3= Szaniewska 
  |first4=Nicolas E.|last4= Gaudin
   | title =A test of the Poincare dodecahedral space topology hypothesis with the WMAP CMB data
  | journal = Astronomy and Astrophysics
  | volume =482
  | issue =3
  | pages =747–753
  | year = 2008
  | arxiv =0801.0006
  | doi =10.1051/0004-6361:20078777
  | id =
| bibcode=2008A&A...482..747L| s2cid =1616362
 }}</ref>
Data analysis from the [[Planck (spacecraft)|Planck spacecraft]] suggests that there is no observable non-trivial topology to the universe.<ref>Planck Collaboration, "[https://arxiv.org/abs/1502.01593 Planck 2015 results. XVIII. Background geometry & topology]", (2015) ArXiv 1502.01593</ref>

==Constructions and examples==

*[[Surgery theory|Surgery]] on a knot in the 3-sphere ''S''<sup>3</sup> with framing +1 or &minus;1 gives a homology sphere.
*More generally, surgery on a link gives a homology sphere whenever the matrix given by intersection numbers (off the diagonal) and framings (on the diagonal) has determinant +1 or &minus;1.
*If ''p'', ''q'', and ''r'' are pairwise relatively prime positive integers then the link of the singularity ''x''<sup>''p''</sup> + ''y''<sup>''q''</sup> + ''z''<sup>''r''</sup> = 0 (in other words, the intersection of a small 3-sphere around 0 with this complex surface) is a [[Brieskorn manifold]] that is a homology 3-sphere, called a [[Egbert Brieskorn|Brieskorn]] 3-sphere Σ(''p'', ''q'', ''r''). It is homeomorphic to the standard 3-sphere if one of ''p'', ''q'', and ''r'' is 1, and Σ(2, 3, 5) is the Poincaré sphere.
*The [[connected sum]] of two oriented homology 3-spheres is a homology 3-sphere. A homology 3-sphere that cannot be written as a connected sum of two homology 3-spheres is called '''irreducible''' or '''prime''', and every homology 3-sphere can be written as a connected sum of prime homology 3-spheres in an essentially unique way.  (See [[Prime decomposition (3-manifold)]].)
*Suppose that <math>a_1, \ldots, a_r</math> are integers all at least 2 such that any two are coprime. Then the [[Seifert fiber space]]

:: <math>\{b, (o_1,0);(a_1,b_1),\dots,(a_r,b_r)\}\,</math>

:over the sphere with exceptional fibers of degrees ''a''<sub>1</sub>, ..., ''a''<sub>''r''</sub> is a homology sphere, where the ''b'''s are chosen so that

:: <math>b+b_1/a_1+\cdots+b_r/a_r=1/(a_1\cdots a_r).</math>

:(There is always a way to choose the ''b''&prime;s, and the homology sphere does not depend (up to isomorphism) on the choice of ''b''&prime;s.) If ''r'' is at most 2 this is just the usual 3-sphere; otherwise they are distinct non-trivial homology spheres. If the ''a''&prime;s are 2, 3, and 5 this gives the Poincaré sphere. If there are at least 3 ''a''&prime;s, not 2, 3, 5, then this is an acyclic  homology 3-sphere with infinite fundamental group  that  has a [[Thurston geometry]] modeled on the universal cover of [[SL2(R)|SL<sub>2</sub>('''R''')]].

==Invariants==
*The [[Rokhlin invariant]] is a <math>\Z/2\Z</math>-valued invariant of homology 3-spheres.
*The [[Casson invariant]] is an integer valued invariant of homology 3-spheres, whose reduction mod 2 is the Rokhlin invariant.

==Applications==
If ''A'' is a homology 3-sphere not homeomorphic to the standard 3-sphere, then the [[suspension (topology)|suspension]] of ''A'' is an example of a 4-dimensional [[homology manifold]] that is not a [[topological manifold]]. The double suspension of ''A'' is homeomorphic to the standard 5-sphere, but its [[triangulation (topology)|triangulation]] (induced by some triangulation of ''A'') is not a [[PL manifold]]. In other words, this gives an example of a finite [[simplicial complex]] that is a topological manifold but not a PL manifold. (It is not a PL manifold because the [[link (geometry)|link]] of a point is not always a 4-sphere.)

Galewski and Stern showed that all compact topological manifolds (without boundary) of dimension at least 5 are homeomorphic to simplicial complexes [[if and only if]] there is a homology 3 sphere Σ with [[Rokhlin invariant]] 1 such that the [[connected sum]] Σ#Σ of Σ with itself bounds a smooth acyclic 4-manifold. [[Ciprian Manolescu]] showed<ref>{{cite journal |last=Manolescu|first=Ciprian |arxiv=1303.2354 |title=Pin(2)-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture | journal=Journal of the American Mathematical Society |volume=29|date=2016|pages=147–176|doi=10.1090/jams829|doi-access=free}}</ref> that there is no such homology sphere with the given property, and therefore, there are 5-manifolds not homeomorphic to simplicial complexes. In particular, the example originally given by Galewski and Stern<ref>{{cite book|first1=David|last1=Galewski|first2=Ronald|last2= Stern|author2-link=Ronald J. Stern|contribution=A universal 5-manifold with respect to simplicial triangulations|title=Geometric Topology (Proceedings Georgia Topology Conference, Athens Georgia, 1977)|year=1979|publisher= [[Academic Press]]|location= New York-London|pages=345–350|mr=0537740}}</ref> is not triangulable.

==See also==
* [[Eilenberg–MacLane space]]
* [[Moore space (algebraic topology)]]

==References==
{{Reflist}}

==Selected reading==
*  {{cite journal|first=Emmanuel|last= Dror|title=Homology spheres|journal=[[Israel Journal of Mathematics]]|volume= 15 |year=1973|issue= 2|pages= 115–129|doi=10.1007/BF02764597|doi-access=|mr=0328926|s2cid= 189796498}}
* {{cite journal|first1=David|last1=Galewski|first2=Ronald|last2= Stern|author2-link=Ronald J. Stern|jstor=1971215    |title=Classification of simplicial triangulations of topological manifolds|journal= [[Annals of Mathematics]] |volume=111 |year=1980|issue= 1|pages=1–34|mr=0558395|doi=10.2307/1971215}}
* [[Robion Kirby]], Martin Scharlemann, ''Eight faces of the Poincaré homology 3-sphere''. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp.&nbsp;113&ndash;146, [[Academic Press]], New York-London, 1979.
* {{cite journal|first=Michel|last= Kervaire|authorlink=Michel Kervaire| jstor=1995269 |title=Smooth homology spheres and their fundamental groups|journal= [[Transactions of the American Mathematical Society]] |volume=144 |year=1969|pages= 67–72|doi= 10.1090/S0002-9947-1969-0253347-3|  mr=0253347|s2cid= 54063849|doi-access=free}}
* Nikolai Saveliev, ''Invariants of Homology 3-Spheres'', Encyclopaedia of Mathematical Sciences, vol 140. Low-Dimensional Topology, I. Springer-Verlag, Berlin, 2002.  {{MathSciNet|1941324}} {{ISBN|3-540-43796-7}}

==External links==
*[http://www.eg-models.de/models/Simplicial_Manifolds/2003.04.001/_preview.html A 16-Vertex Triangulation of the Poincaré Homology 3-Sphere and Non-PL Spheres with Few Vertices] by [[Anders Björner]] and [[Frank H. Lutz]]
*Lecture by [[David Gillman]] on [http://media.pims.math.ca/realvideo-ram/science/2002/cascade/gillman/gillman.ram The best picture of Poincare's homology sphere ]

[[Category:Topological spaces]]
[[Category:Homology theory]]
[[Category:3-manifolds]]
[[Category:Spheres]]